Problem: Select all polynomials that are divisible by $(x-1)$. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=3x^3+2x^2-x$ (Choice B) B $B(x)=5x^3-4x^2-x$ (Choice C) C $C(x)=2x^3-3x^2+2x-1$ (Choice D) D $D(x)=x^3+2x^2+3x+2$
The following statements are equivalent: $(x-1)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x-1)$ The remainder of $\dfrac{p(x)}{x-1}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x-{1})$ is equal to $p({1})$. So to check each polynomial is divisible by $(x-1)$, we need to check if that polynomial's value at ${x=1}$ is zero. $\begin{aligned} A({1})&=4 \\\\ B({1})&=0 \\\\ C({1})&=0 \\\\ D({1})&=8 \end{aligned}$ In conclusion, the following polynomials are divisible by $(x-1)$ : $B(x)=5x^3-4x^2-x$ $C(x)=2x^3-3x^2+2x-1$